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5.3 二次曲线的切线

5.3 二次曲线的切线. 定义 5.3.1 如果直线与二次曲线相交于相互重合的两个点，那么这条直线就叫做二次曲线的 切线 ，这个重合的交点叫做 切点 ，如果直线全部在二次曲线上，我们也称它为二次曲线的 切线 ，直线上的每个点都可以看作 切点. 定义 5.3.2 二次曲线 (1) 上满足条件 F 1 ( x 0 , y 0 )= F 2 ( x 0 , y 0 )=0 的点 ( x 0 , y 0 ) 叫做二次曲线的奇异点，简称奇点；二次曲线的非奇异点叫做二次曲线的正常点.

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5.3 二次曲线的切线

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  1. 5.3二次曲线的切线 定义5.3.1如果直线与二次曲线相交于相互重合的两个点，那么这条直线就叫做二次曲线的切线，这个重合的交点叫做切点，如果直线全部在二次曲线上，我们也称它为二次曲线的切线，直线上的每个点都可以看作切点. 定义5.3.2二次曲线(1)上满足条件F1(x0,y0)= F2(x0,y0)=0的点(x0,y0)叫做二次曲线的奇异点，简称奇点；二次曲线的非奇异点叫做二次曲线的正常点.

  2. 定理5.3.1如果(x0,y0)是二次曲线(1)的正常点，那么通过(x0,y0)的切线方程是 (x-x0)F1 (x0,y0)+ (y-y0)F2 (x0,y0)=0， (x0,y0)是它的切点. 如果(x0,y0)是二次曲线(1)的奇异点，那么通过(x0,y0)的切线不确定，或者说过点(x0,y0)的每一条直线都是二次曲线(1)的切线. 推论 如果(x0,y0)是二次曲线(1)的正常点，那么通过(x0,y0)的切线方程是：

  3. 例1 求二次曲线x2-xy+y2+2x-4y-3=0在点(2,1)的切线方程 解：因为F(2,1)=4-2+1+4-4-3=0, 且 F1(2,1)=5/2≠0, F2(2,1)=-2 ≠0 所以(2,1)是二次曲线上的正常点，因此得在 点(2,1)的切线方程为： 5/2 (x-2)-2(y-1)=0 即：5x-4y-6=0

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